The bound names the hottest part and the cost does not

A rule that is never right and cannot be far wrong measures what moving in the hottest component costs against playing the whole board correctly. Over 440 lines it costs something on 17, it never exceeds the largest temperature on the board, and the same experiment with the ordering reversed breaks that bound on 54 — which is what makes the bound a claim rather than an observation.

The bound is exactly right and it is about the wrong quantity. Sweeping four pools and boards of two, three and four parts — 4,240 lines — the cost turns out to be nothing, half a point, or one point, and nothing else, on boards whose largest temperature reaches three.

The bound is attained, and it is nowhere near the cost #

The bound is tight in the only sense a bound can be. On 100 of the 4,240 lines the cost equals the largest temperature exactly, so no theorem of the form at most c times the largest temperature holds with c below one. Whoever proved this proved the best available statement about the hottest component.

And the worst cost anywhere is one point, on boards whose largest temperature is three. The two facts are compatible because the lines that attain the bound are lines where the largest temperature is itself small — where everything on the board is cool, the bound is one point and one point is what gets paid. Where the board is hot, the bound is three and the cost is nought or one.

So the bound is doing two different things at once. On a cool board it is a description; on a hot board it is a permission the rule never takes up. A reader handed the theorem and a board with a three-point fight on it would budget three points for playing greedily, and the budget is wrong by a factor of three at least.

Two values, and nothing between #

The theorem states an interval and a reader reads an interval: a cost somewhere between nothing and the largest temperature, presumably spread out, most of it small. It is not spread out at all. The cost is a set of two values with a large gap above it, and nothing at all between a half and one.

That is not an artefact of whole-number pools, which is why there are four of them. Two are built from whole numbers and the cost on those is nought or one. One is built so that every component’s temperature has a half in it, and the cost on that pool is nought or a half — never one, and never a quarter. The fourth mixes them and produces both a half and a one.

The cost is the size of the smallest step the board’s values can take. In a pool of whole-number switches the values move in whole points and a misplay costs a point; in a pool of halves it costs a half. What the rule loses is not an amount of temperature — it is one unit of whatever granularity the position is built at, and the bound has no way of saying so because it is measured in temperature.

That is also why a bound of this shape cannot be improved. A theorem saying the cost is at most the smallest step the values take would be false on a board of huge switches and tiny steps, and a theorem saying at most the largest temperature is what survives both kinds of board. The price of a statement that covers everything is a statement that describes nothing.

The quantity the theorem does not mention #

Compare the cost with the coolest part of the board rather than the hottest, and the picture changes completely. On 376 of the 389 lines that cost anything, the cost is at most the coolest component’s temperature, and on the remaining 13 it is at most twice it. The worst ratio over all 4,240 lines is exactly 2.

That is a much tighter statement than the theorem and it is a conjecture rather than a theorem. It is also the right shape of statement, because it explains the granularity: the coolest fight on the board is the last one played, and the last fight is where a misordering is paid for. Playing the hottest first cannot go wrong while the hot fights are still there to be taken; it goes wrong at the end, over something small, and the something small is the coolest component.

The rule’s error is an endgame error, and the bound is stated in the currency of the opening. That is the same split a rule with a guarantee records between what the temperature promises and where the promise is spent, and a rule with no promise at all is the control for it: a rule with no bound behind it goes wrong by amounts that do scale with the board.

Where the 13 come from #

The 13 lines that cost more than the coolest part are worth knowing, because they say what the conjecture would have to survive.

All 13 are boards of four parts, and all 13 have exactly two components at the coolest temperature — counted rather than noticed, since a single exception with three would change what the conjecture says. A board of ⟨5|1⟩, two copies of ⟨1|0⟩ and ⟨6|⟨3|1⟩⟩ has a coolest temperature of a half and costs a full point; so do the other twelve, with the same shape.

The mechanism is visible once the shape is. A single cool component is one last fight and costs at most itself. Two identical cool components are two last fights, and the greedy rule can misorder them against the hotter part in a way that pays for both — half a point twice. So the conjecture is not at most the coolest temperature; it is at most that times the number of copies, and on these boards the number is two.

Whether it stays two on a board with three copies is not measured here and it is the obvious thing to measure next: the sweep runs to four parts, and a board with three copies of the coolest component needs at least four parts to be interesting.

What happens as the board grows #

The sweep runs boards of two, three and four parts from each pool, which is enough to watch one thing move and one thing stay still.

The share of lines that cost anything rises with the board. On the sente-and-gote pool it is 15 of 72 at two parts, 33 of 240 at three and 134 of 660 at four — roughly a fifth, an eighth, a fifth. On the whole-number pool it is 6 of 110, 17 of 440 and 75 of 1,430. The direction is up once the counting is done per line, and the reason is plain: more parts means more orderings, and more orderings means more chances to take them in the wrong one.

The worst cost does not move at all. It is one point on every pool with whole numbers in it and half a point on the pool without, at two parts, at three and at four. A board with four fights on it gives the greedy rule four opportunities to go wrong and it still loses one step in total, not four.

That is the fact behind the conjecture and it is the surprising half of the measurement. A reader would expect a rule that errs locally to err repeatedly, so that a longer game costs more — and what actually happens is that the errors do not accumulate. Playing the hottest first means the board is always being reduced from the top, and a misordering at one point is corrected by the ordering that follows it; only the very last pair of fights has nothing after it to do the correcting. That is why the cost is one step and why it is the coolest step, and it is also why the 13 exceptions need two components at the bottom rather than one.

What a bound of this kind is for #

It is worth being clear that none of this is a complaint about the theorem.

What is at stake is what the temperature measures, and a bound has to hold on every board, including the ones nobody would build. The pools here are finite and chosen; a pool with a component of temperature 100 and steps of a thousandth would cost the rule a thousandth and would leave the bound at 100, and a pool of enormous steps and cool components would push the cost up to the bound. Both exist, so the theorem cannot mention either quantity alone.

What the measurement adds is the shape of the gap. A user of the rule wants to know what it will cost them, and the theorem answers with a number that is up to three times too large and is, on the lines where it is exact, exact for a reason that has nothing to do with heat. Knowing that the cost is one step of the board’s own granularity, paid at the end, is a usable fact; knowing that it is at most the largest temperature is not.

Big is not the same as hot makes the neighbouring point about the rule — that a player sizing moves by what changes hands ranks components differently from one sizing them by what is at stake. This is the same distinction one level up, about the bound: the quantity a theorem is stated in and the quantity the behaviour is governed by need not be the same quantity, and a tight theorem is no evidence that they are.

What the sweep cannot say #

Four pools and three sizes is a sample of boards, not a proof. The two-value distribution is a fact about 4,240 lines. A pool built to produce a cost of three quarters would presumably produce one, and nothing here says it cannot.

The conjecture about the coolest part is a conjecture. It survives 4,240 lines with a worst ratio of exactly 2, which is a strong thing for a guess to do, and it is not a proof and the 13 exceptions are the place a proof would have to start.

And every board here has at most four parts. A real endgame has a dozen, and the rule’s cost on a dozen parts is not measured — the optimal play a cost has to be measured against is a minimax over the whole sum, which is what a value costs to compute and is what stops the sweep at four. An environment instead of a stack is what a real endgame is modelled with when the parts are too many to enumerate, and it is a different measurement from this one.

The convention the cost is measured under #

Scores, not outcomes. Every component here is a switch or a switch with a fight underneath, every board has a value that is a number once the fights are settled, and the cost of a line is the difference between the score the rule reaches and the score best play reaches. Left wants the total large and Right wants it small, so the cost is signed and the sign is taken care of before anything is counted.

Both players play the rule under test. That is the convention playing the hottest states and it matters here, because a measurement where one side played greedily and the other optimally would be measuring something else — how much a good player takes off a greedy one, rather than what the rule costs the person using it.

And the largest temperature is computed from the board at the start. A bound that read the temperature at each turn would be a different and smaller quantity, since the board cools as it is played; the theorem is stated at the start and the cost is compared with it there.

The surprise: a tight bound can still be about the wrong thing #

The usual reason a bound is uninformative is that it is loose — attained nowhere, provable only because the proof is crude. The natural response is to look for a tighter one.

This bound is attained. One hundred lines pay exactly the largest temperature, so nothing of the form at most c times the largest temperature does better. By the ordinary test, the theorem is as good as a theorem about that quantity can be.

And the cost is still not described by it, because the lines that attain it are not the lines anybody was worrying about. They are the boards where everything is cool, where the bound is small and the cost is one step and the two coincide. On the boards with a real fight on them — where a reader would actually want to know what greed costs — the bound is three and the answer is nought or one.

Attainment says the bound cannot be improved in its own terms. It says nothing about whether its terms are the right ones. A bound attained only on the degenerate end of its range is a bound pinned there by cases the user does not care about, and the way to find that out is to measure the whole distribution rather than the worst case — which is the one thing a proof never does.

That is worth a rule of thumb, because the situation is common: a theorem with an interval in it, a measurement with a distribution, and a reader who has only the theorem. Ask what the worst cases have in common before believing the worst case is the case. Here they have in common that the board is uniformly cool, which is the one shape the rule cannot be punished on and the one shape a reader would never think to worry about.

There is a practical test in that, and it is cheap. Take the lines where a bound is attained and ask what they have in common; if they all sit at one end of some quantity the bound does not mention, the bound is being held there by that quantity rather than by the one it names. Here the attaining lines are the lines where the largest and smallest temperatures coincide — the pool where every temperature is a half attains the bound on every single line that costs anything, 24 of 24 across the three board sizes — and on the three pools where the two come apart, attainment is 76 lines of 365. The bound is exact exactly where it has nothing to be exact about.

Still open: how many copies of the coolest part it takes #

The conjecture the measurement suggests is that the cost is at most the coolest temperature on the board, times the number of components that share it. On the 4,240 lines here that number never exceeds two, and every line that exceeds the coolest temperature has exactly two copies of the cheapest fight.

The measurement that would test it is a pool with three components of the same small temperature and a board of five or six parts, played out against optimal play the same way. If the cost reaches three times the coolest temperature there, the conjecture is a scale rather than a constant and the bound to look for is a sum over the cool end of the board. If it stays at two, then two is a fact that wants explaining, and the explanation would be about how many times a greedy player can be made to answer the wrong small fight before the board runs out.